Homotopy
The Stiefel manifolds fit into a family of fibrations, thus the first non-trivial homotopy group of the space Vk(Rn) is in dimension n - k. Moreover, if n - k ∈ 2Z or if k = 1. if n - k is odd and k > 1. This result is used in the obstruction-theoretic definition of Stiefel-Whitney classes.
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